In a game we keep throwing a die until it shows a three.
1: If the probability to get a three is $p$, what is the probability for the game to stop sometime?
My solution:
The probability $P_s$ for the game to stop equals the sum of the probabilities for the game to stop at different rounds, i.e $P_s = P(\textrm{game stops at round 1}) + P(\textrm{game stops at round 2})+ \ldots = p+p^2+p^3+p^4+\ldots$ which converges to $\lim_{n->\infty} n(p+p^n)/2$.
However, my professor says that $P_s= 1$ for every $p \neq 0$. Why is that?
2: My professor also says that the probability for the game to stop after finite number of rounds is a totally different problem, although they share same final answer.
Why would this be? Is there a difference between stopping sometime and stopping after finite number of rounds?
1: Your solution to the first question goes wrong in calculating the sum of the probabilities that the game ends at the different possible rounds. Note that the game ending at round 3, for example, presupposes that the game didn't end at round 1 or 2. And the probability that the game does not end when you throw the die is $1-p$. Since every roll of the die is independent:
$$ P(\textrm{game stops at round 3}) = (1-p)^2p\,.$$
Let $P_s$ be the probability that the game eventually stops. In line with the above:
$$ P_s = P(\textrm{game stops at some round}) = p + p(1-p) + p(1-p)^2 + \ldots $$
If $p=0$, every term on the right hand side is zero, so $P_s = 0$. If $0 < p \leq 1$, use the formula for the sum of a convergent geometric series to obtain
$$P_s = \frac{p}{1-(1-p)} = \frac{p}{p} = 1\,.$$
Thus, your professor is right on this one.
2: The event that the game stops at some time is exactly the same thing as the event that the game stops after a finite number of rounds, since there does not exist any "round infinity". If the game doesn't stop at a finite time, it doesn't stop at all.
So the task "Calculate the probability that the game stops after a finite number of rounds" is exactly the same problem that we solved above. The slightly different phrasing makes no difference. Either your professor and you have misunderstood each other, or your professor is wrong about the second point.